A decomposition theorem for higher rank Coxeter groups

In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups.Comment: 7 pages, 3 figure

Bridge number and tangle products

We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bridge number of a tangle product is at least the sum of the bridge numbers of the two factor links up to a constant error.Comment: 13 pages, 11 figure

Alternating Augmentations of Links

We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating.Comment: 5 pages, 7 figure

Idempotents in Tangle Categories Split

In this paper we use 3-manifold techniques to illuminate the structure of the category of tangles. In particular, we show that every idempotent morphism $A$ in such a category naturally splits as $A=B\circ C$ such that $C\circ B$ is an identity morphism.Comment: 10 pages, 5 figure

Companions of the unknot and width additivity

It has been conjectured that for knots $K$ and $K'$ in $S^3$, w(K#K')= w(K)+w(K')-2. Scharlemann and Thompson have proposed potential counterexamples to this conjecture. For every $n$, they proposed a family of knots ${K^n_i}$ for which they conjectured that w(B^n#K^n_i)=w(K^n_i) where $B^n$ is a bridge number $n$ knot. We show that for $n>2$ none of the knots in ${K^n_i}$ produces such counterexamples.Comment: 12 pages, 11 figure

Knots with compressible thin levels

We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.Comment: 24 pages, 6 figure

High Distance Bridge Surfaces

Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Sigma for (M,L) of genus g that intersects L in 2b points and has distance at least n. More generally, given two possibly disconnected surfaces S and S', each with some even number (possibly zero) of marked points, and integers b, c, g, and n, we construct a compact, orientable manifold M with boundary S \cup S' such that M contains a c-component tangle T with a bridge surface Sigma of genus g that separates the boundary of M into S and S', |T \cap Sigma|=2b and T intersects S and S' exactly in their marked points, and Sigma has distance at least n.Comment: 17 pages, 13 figures; v2 clarifying revisions made based on referee's comment

Dynamics of Embedded Curves by Doubly-Nonlocal Reaction-Diffusion Systems

We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of complexity for embedded curves. This line of investigation culminates in a family of complexity bounds that relate a rather broad class of models to a generalized, or weighted, variant of the crossing number. Our dynamic results include global well-posedness of the associated partial differential equations, regularity of equilibria for these flows as well as a more detailed investigation of dynamics near such equilibria. Finally, we explore a few global dynamical properties of these models numerically.Comment: 49 pages, 3 figure

A prime decomposition theorem for the 2-string link monoid

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links to commute under the stacking operation.Comment: Various additions and modifications, mostly suggested by the referee. Now 27 pages, 11 figures. To appear in J. Knot Theory Rami

The incompatibility of crossing number and bridge number for knot diagrams

We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.Comment: 14 pages, 13 figure